Concurrency Theory(WS 2010/11) Out: Thu, Dec 16 Due: Mon, Dec 20
Exercise Sheet 8
Jun.-Prof. Roland Meyer, Georgel C˘alin Technische Universit¨at Kaiserslautern
Problem 1: Reachability of Upward-Closed Sets
Consider wsts(Γ,→, γ0,≤). Letprej(I) := pre(. . . pre
| {z }
jtimes
(I). . .)for upward closed setI ⊆Γ.
(a) Show thatIj =Sj
l=0prel(I)withIj as it has been defined in the lecture.
(b) Prove thatI is reachable fromγ in≤nsteps if and only ifγ ∈In.
Problem 2+3: Generalized Lossy Channel Systems
Consider the following variation of a lcs: assume one of the symbolss ∈ M can not be lost during send/receive by any channel but that a channel can contain at mostk ∈ N symbolss.
A transition that wants to send thek + 1st symbolsis blocked. Such a generalized lcs can be represented by a standard lcs using as states the Cartesian productQ×{0, . . . , k}whereQis the set of states of the original system. The resulting lcs transitions are schematically represented below (for0≤i < k).
(q1, i) (q2, i+ 1) c!s
You are asked to give an implementation of(q1, i)−→(qc!s 2, i+ 1)by several lossy transiti- ons. Your model should check that preciselyi symbols s are present in the channel cbefore appending the extras. Hint 1: TakeM∪ {#}as the alphabet of the resulting lcs. Hint 2: What happens if you usec?mand afterwardsc!mform∈M ∪ {#}. The solution needs a notion of round.
Problem 4: Coverability for Lossy Channel Systems
q0 N!1 q1 q2 q3 q4
A!1 N!0
A?0
N?1 N!0
N?0
Determine if the configurations(q4, 0
ε
)and(q4, ε
1
)(where the upper channel entry is for N and the lower for A) are coverable by specifying the minimal predecessors created by the backward coverability procedure discussed in class.
